Every p-subgroup is well-placed in some Sylow subgroup
Statement that involves choosing a Sylow subgroup
Suppose is a finite group, is a prime number, and is a Conjugacy functor (?) on the -subgroups of . Suppose is a -subgroup of . Then, there exists a -Sylow subgroup of such that is a Well-placed subgroup (?) in relative to the conjugacy functor .
Statement that involves conjugating the subgroup
Suppose is a finite group, is a prime number, and is a Conjugacy functor (?) on the -subgroups of . Suppose is a -subgroup of contained in a -Sylow subgroup . Then, there exists a subgroup of , that is well-placed in , and is conjugate to in .
- Every p-subgroup is conjugate to a p-subgroup whose normalizer in the Sylow is Sylow in its normalizer
- Conjugacy functor sends every subgroup to a normalizer-relatively normal subgroup: For any -subgroup , .
- Sylow subgroups exist
- Sylow implies order-dominating
Given: A finite group , a prime , a conjugacy functor on the -subgroups. A -subgroup of .
To prove: There exists a -Sylow subgroup of such that is well-placed in relative to .
Proof: Define the following ascending chain of subgroups . , and is a -Sylow subgroup of . By fact (1), this chain of subgroups is an ascending chain of -subgroups. Let be a -Sylow subgroup of containing the union of the s (such a exists by facts (2) and (3)). We now prove that is well-placed in . This follows easily from the definition.